Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.8s

Alternatives: 11

Speedup: N/A×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{1}{\frac{s}{\pi}}}}} + -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (/ s PI)))))))
     -1.0)))))

C

float code(float u, float s) {
	return s * -logf(((1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((1.0f / (s / ((float) M_PI)))))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(1.0) / Float32(s / Float32(pi)))))))) + Float32(-1.0)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(1.0) / (s / single(pi)))))))) + single(-1.0)));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\frac{1}{\frac{s}{\pi}}}}}} + -1\right) \]

   2. inv-pow 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\frac{s}{\pi}\right)}^{-1}}}}} + -1\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\frac{s}{\pi}\right)}^{-1}}}}} + -1\right) \]
7. Step-by-step derivation

   1. unpow-1 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\frac{1}{\frac{s}{\pi}}}}}} + -1\right) \]

8. Simplified 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\frac{1}{\frac{s}{\pi}}}}}} + -1\right) \]

9. Final simplification 99.0%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{1}{\frac{s}{\pi}}}}} + -1\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Final simplification 99.0%

   \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]
6. Add Preprocessing

Alternative 3: 25.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}, 1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma -4.0 (/ (+ (* 0.5 (* u PI)) (* PI -0.25)) s) 1.0))))

C

float code(float u, float s) {
	return -s * logf(fmaf(-4.0f, (((0.5f * (u * ((float) M_PI))) + (((float) M_PI) * -0.25f)) / s), 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(-4.0), Float32(Float32(Float32(Float32(0.5) * Float32(u * Float32(pi))) + Float32(Float32(pi) * Float32(-0.25))) / s), Float32(1.0))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
6. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s} + 1\right)} \]

   2. fma-define 25.1%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}, 1\right)\right)} \]

7. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}, 1\right)\right)} \]

8. Final simplification 25.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}, 1\right)\right) \]
9. Add Preprocessing

Alternative 4: 25.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    1.0
    (* 4.0 (/ (- (* -0.25 (* u PI)) (+ (* PI -0.25) (* (* u PI) 0.25))) s))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * (((-0.25f * (u * ((float) M_PI))) - ((((float) M_PI) * -0.25f) + ((u * ((float) M_PI)) * 0.25f))) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(Float32(Float32(-0.25) * Float32(u * Float32(pi))) - Float32(Float32(Float32(pi) * Float32(-0.25)) + Float32(Float32(u * Float32(pi)) * Float32(0.25)))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (((single(-0.25) * (u * single(pi))) - ((single(pi) * single(-0.25)) + ((u * single(pi)) * single(0.25)))) / s))));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]

6. Final simplification 25.1%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right)}{s}\right) \]
7. Add Preprocessing

Alternative 5: 14.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{-s} \cdot \left(-4 \cdot \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (/ (pow s 2.0) (- s)) (* -4.0 (/ (+ (* 0.5 (* u PI)) (* PI -0.25)) s))))

C

float code(float u, float s) {
	return (powf(s, 2.0f) / -s) * (-4.0f * (((0.5f * (u * ((float) M_PI))) + (((float) M_PI) * -0.25f)) / s));
}

Julia

function code(u, s)
	return Float32(Float32((s ^ Float32(2.0)) / Float32(-s)) * Float32(Float32(-4.0) * Float32(Float32(Float32(Float32(0.5) * Float32(u * Float32(pi))) + Float32(Float32(pi) * Float32(-0.25))) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = ((s ^ single(2.0)) / -s) * (single(-4.0) * (((single(0.5) * (u * single(pi))) + (single(pi) * single(-0.25))) / s));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate--r+ 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi}}{s}\right) \]

   2. cancel-sign-sub-inv 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi}}{s}\right) \]

   3. distribute-rgt-out-- 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi}{s}\right) \]

   4. *-commutative 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(\pi \cdot u\right)} \cdot \left(0.25 - -0.25\right) + \left(-0.25\right) \cdot \pi}{s}\right) \]

   5. metadata-eval 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}\right) \]

   6. metadata-eval 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}\right) \]

   7. *-commutative 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)} \]
8. Step-by-step derivation

   1. neg-sub0 11.9%

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   2. flip-- 14.5%

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   3. metadata-eval 14.5%

      \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   4. pow2 14.5%

      \[\leadsto \frac{0 - \color{blue}{{s}^{2}}}{0 + s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   5. add-sqr-sqrt 14.5%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   6. sqrt-unprod 9.0%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   7. sqr-neg 9.0%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   8. sqrt-unprod -0.0%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   9. add-sqr-sqrt 7.7%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   10. sub-neg 7.7%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{0 - s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   11. neg-sub0 7.7%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{-s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   13. sqrt-unprod 9.0%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   14. sqr-neg 9.0%

       \[\leadsto \frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   15. sqrt-unprod 14.5%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   16. add-sqr-sqrt 14.5%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

9. Applied egg-rr 14.5%

   \[\leadsto \color{blue}{\frac{0 - {s}^{2}}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]
10. Step-by-step derivation

    1. sub0-neg 14.5%

       \[\leadsto \frac{\color{blue}{-{s}^{2}}}{s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

11. Simplified 14.5%

    \[\leadsto \color{blue}{\frac{-{s}^{2}}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

12. Final simplification 14.5%

    \[\leadsto \frac{{s}^{2}}{-s} \cdot \left(-4 \cdot \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}\right) \]
13. Add Preprocessing

Alternative 6: 13.2% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \left(-1 + \left(1 - s\right)\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(0.25 + u \cdot -0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (+ -1.0 (- 1.0 s))
  (* 4.0 (/ (+ (* PI (+ 0.25 (* u -0.25))) (* PI (* u -0.25))) s))))

C

float code(float u, float s) {
	return (-1.0f + (1.0f - s)) * (4.0f * (((((float) M_PI) * (0.25f + (u * -0.25f))) + (((float) M_PI) * (u * -0.25f))) / s));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(-1.0) + Float32(Float32(1.0) - s)) * Float32(Float32(4.0) * Float32(Float32(Float32(Float32(pi) * Float32(Float32(0.25) + Float32(u * Float32(-0.25)))) + Float32(Float32(pi) * Float32(u * Float32(-0.25)))) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(-1.0) + (single(1.0) - s)) * (single(4.0) * (((single(pi) * (single(0.25) + (u * single(-0.25)))) + (single(pi) * (u * single(-0.25)))) / s));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

      \[\leadsto \left(-\color{blue}{\sqrt{s} \cdot \sqrt{s}}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. distribute-rgt-neg-in 97.8%

      \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
7. Step-by-step derivation

   1. distribute-rgt-neg-out 97.8%

      \[\leadsto \color{blue}{\left(-\sqrt{s} \cdot \sqrt{s}\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. add-sqr-sqrt 99.0%

      \[\leadsto \left(-\color{blue}{s}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. expm1-log1p-u 99.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-s\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. expm1-undefine 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

8. Applied egg-rr 24.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
9. Step-by-step derivation

   1. sub-neg 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} + \left(-1\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. log1p-undefine 24.5%

      \[\leadsto \left(e^{\color{blue}{\log \left(1 + \left(-s\right)\right)}} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. rem-exp-log 24.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(-s\right)\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. unsub-neg 24.5%

      \[\leadsto \left(\color{blue}{\left(1 - s\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   5. metadata-eval 24.5%

      \[\leadsto \left(\left(1 - s\right) + \color{blue}{-1}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

10. Simplified 24.5%

    \[\leadsto \color{blue}{\left(\left(1 - s\right) + -1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

11. Taylor expanded in s around -inf 13.5%

    \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
12. Step-by-step derivation

    1. associate--r+ 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) - 0.25 \cdot \left(u \cdot \pi\right)}}{s}\right) \]

    2. cancel-sign-sub-inv 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)}}{s}\right) \]

    3. cancel-sign-sub-inv 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) + \left(--0.25\right) \cdot \pi\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)}{s}\right) \]

    4. metadata-eval 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\left(-0.25 \cdot \left(u \cdot \pi\right) + \color{blue}{0.25} \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)}{s}\right) \]

    5. associate-*r* 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\left(\color{blue}{\left(-0.25 \cdot u\right) \cdot \pi} + 0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)}{s}\right) \]

    6. distribute-rgt-out 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\color{blue}{\pi \cdot \left(-0.25 \cdot u + 0.25\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)}{s}\right) \]

    7. metadata-eval 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \color{blue}{-0.25} \cdot \left(u \cdot \pi\right)}{s}\right) \]

    8. *-commutative 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \color{blue}{\left(u \cdot \pi\right) \cdot -0.25}}{s}\right) \]

    9. *-commutative 13.5%

       \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \color{blue}{\left(\pi \cdot u\right)} \cdot -0.25}{s}\right) \]

    10. associate-*l* 13.5%

        \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \color{blue}{\pi \cdot \left(u \cdot -0.25\right)}}{s}\right) \]

13. Simplified 13.5%

    \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \color{blue}{\left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}\right)} \]

14. Final simplification 13.5%

    \[\leadsto \left(-1 + \left(1 - s\right)\right) \cdot \left(4 \cdot \frac{\pi \cdot \left(0.25 + u \cdot -0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}\right) \]
15. Add Preprocessing

Alternative 7: 13.2% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \left(-4 \cdot \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}\right) \cdot \left(-1 + \left(1 - s\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (* -4.0 (/ (+ (* 0.5 (* u PI)) (* PI -0.25)) s)) (+ -1.0 (- 1.0 s))))

C

float code(float u, float s) {
	return (-4.0f * (((0.5f * (u * ((float) M_PI))) + (((float) M_PI) * -0.25f)) / s)) * (-1.0f + (1.0f - s));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(-4.0) * Float32(Float32(Float32(Float32(0.5) * Float32(u * Float32(pi))) + Float32(Float32(pi) * Float32(-0.25))) / s)) * Float32(Float32(-1.0) + Float32(Float32(1.0) - s)))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(-4.0) * (((single(0.5) * (u * single(pi))) + (single(pi) * single(-0.25))) / s)) * (single(-1.0) + (single(1.0) - s));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

      \[\leadsto \left(-\color{blue}{\sqrt{s} \cdot \sqrt{s}}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. distribute-rgt-neg-in 97.8%

      \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
7. Step-by-step derivation

   1. distribute-rgt-neg-out 97.8%

      \[\leadsto \color{blue}{\left(-\sqrt{s} \cdot \sqrt{s}\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. add-sqr-sqrt 99.0%

      \[\leadsto \left(-\color{blue}{s}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. expm1-log1p-u 99.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-s\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. expm1-undefine 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

8. Applied egg-rr 24.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
9. Step-by-step derivation

   1. sub-neg 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} + \left(-1\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. log1p-undefine 24.5%

      \[\leadsto \left(e^{\color{blue}{\log \left(1 + \left(-s\right)\right)}} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. rem-exp-log 24.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(-s\right)\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. unsub-neg 24.5%

      \[\leadsto \left(\color{blue}{\left(1 - s\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   5. metadata-eval 24.5%

      \[\leadsto \left(\left(1 - s\right) + \color{blue}{-1}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

10. Simplified 24.5%

    \[\leadsto \color{blue}{\left(\left(1 - s\right) + -1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

11. Taylor expanded in s around inf 13.5%

    \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \color{blue}{\left(-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
12. Step-by-step derivation

    1. associate--r+ 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi}}{s}\right) \]

    2. cancel-sign-sub-inv 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi}}{s}\right) \]

    3. distribute-rgt-out-- 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi}{s}\right) \]

    4. *-commutative 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(\pi \cdot u\right)} \cdot \left(0.25 - -0.25\right) + \left(-0.25\right) \cdot \pi}{s}\right) \]

    5. metadata-eval 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}\right) \]

    6. metadata-eval 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}\right) \]

    7. *-commutative 11.9%

       \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}\right) \]

13. Simplified 13.5%

    \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \color{blue}{\left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)} \]

14. Final simplification 13.5%

    \[\leadsto \left(-4 \cdot \frac{0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot -0.25}{s}\right) \cdot \left(-1 + \left(1 - s\right)\right) \]
15. Add Preprocessing

Alternative 8: 13.1% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{s} \cdot \left(-1 + \left(1 - s\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (/ PI s) (+ -1.0 (- 1.0 s))))

C

float code(float u, float s) {
	return (((float) M_PI) / s) * (-1.0f + (1.0f - s));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(pi) / s) * Float32(Float32(-1.0) + Float32(Float32(1.0) - s)))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * (single(-1.0) + (single(1.0) - s));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

      \[\leadsto \left(-\color{blue}{\sqrt{s} \cdot \sqrt{s}}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. distribute-rgt-neg-in 97.8%

      \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\left(\sqrt{s} \cdot \left(-\sqrt{s}\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
7. Step-by-step derivation

   1. distribute-rgt-neg-out 97.8%

      \[\leadsto \color{blue}{\left(-\sqrt{s} \cdot \sqrt{s}\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. add-sqr-sqrt 99.0%

      \[\leadsto \left(-\color{blue}{s}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. expm1-log1p-u 99.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-s\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. expm1-undefine 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

8. Applied egg-rr 24.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
9. Step-by-step derivation

   1. sub-neg 24.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} + \left(-1\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   2. log1p-undefine 24.5%

      \[\leadsto \left(e^{\color{blue}{\log \left(1 + \left(-s\right)\right)}} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   3. rem-exp-log 24.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(-s\right)\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   4. unsub-neg 24.5%

      \[\leadsto \left(\color{blue}{\left(1 - s\right)} + \left(-1\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

   5. metadata-eval 24.5%

      \[\leadsto \left(\left(1 - s\right) + \color{blue}{-1}\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

10. Simplified 24.5%

    \[\leadsto \color{blue}{\left(\left(1 - s\right) + -1\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

11. Taylor expanded in u around 0 13.4%

    \[\leadsto \left(\left(1 - s\right) + -1\right) \cdot \color{blue}{\frac{\pi}{s}} \]

12. Final simplification 13.4%

    \[\leadsto \frac{\pi}{s} \cdot \left(-1 + \left(1 - s\right)\right) \]
13. Add Preprocessing

Alternative 9: 11.8% accurate, 61.9× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(-1 + u \cdot 2\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* PI (+ -1.0 (* u 2.0))))

C

float code(float u, float s) {
	return ((float) M_PI) * (-1.0f + (u * 2.0f));
}

Julia

function code(u, s)
	return Float32(Float32(pi) * Float32(Float32(-1.0) + Float32(u * Float32(2.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate--r+ 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi}}{s}\right) \]

   2. cancel-sign-sub-inv 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi}}{s}\right) \]

   3. distribute-rgt-out-- 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi}{s}\right) \]

   4. *-commutative 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\color{blue}{\left(\pi \cdot u\right)} \cdot \left(0.25 - -0.25\right) + \left(-0.25\right) \cdot \pi}{s}\right) \]

   5. metadata-eval 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}\right) \]

   6. metadata-eval 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}\right) \]

   7. *-commutative 11.9%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)} \]

8. Taylor expanded in u around 0 11.9%

   \[\leadsto \color{blue}{-1 \cdot \pi + 2 \cdot \left(u \cdot \pi\right)} \]
9. Step-by-step derivation

   1. associate-*r* 11.9%

      \[\leadsto -1 \cdot \pi + \color{blue}{\left(2 \cdot u\right) \cdot \pi} \]

   2. distribute-rgt-out 11.9%

      \[\leadsto \color{blue}{\pi \cdot \left(-1 + 2 \cdot u\right)} \]

10. Simplified 11.9%

    \[\leadsto \color{blue}{\pi \cdot \left(-1 + 2 \cdot u\right)} \]

11. Final simplification 11.9%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
12. Add Preprocessing

Alternative 10: 11.6% accurate, 216.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

function code(u, s)
	return Float32(-Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 11.6%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
6. Step-by-step derivation

   1. neg-mul-1 11.6%

      \[\leadsto \color{blue}{-\pi} \]

7. Simplified 11.6%

   \[\leadsto \color{blue}{-\pi} \]
8. Add Preprocessing

Alternative 11: 10.2% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 10.3%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]

6. Taylor expanded in s around 0 10.3%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))